Skip to main content
SpringerLink
Log in

On Generalized Statistical and Ideal Convergence of Metric-Valued Sequences

  • Published:
Ukrainian Mathematical Journal Aims and scope

We consider the notion of generalized density, namely, the natural density of weight g recently introduced in [M. Balcerzak, P. Das, M. Filipczak, and J. Swaczyna, Acta Math. Hung., 147, No. 1, 97–115 (2015)] and primarily study some sufficient and almost converse necessary conditions for the generalized statistically convergent sequence under which the subsequence is also generalized statistically convergent. Some results are also obtained in a more general form by using the notion of ideals. The entire investigation is performed in the setting of general metric spaces extending the recent results of M. Kücükaslan, U. Deger, and O. Dovgoshey, [Ukr. Math. J., 66, No. 5, 712–720 (2014)].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Balcerzak, P. Das, M. Filipczak, and J. Swaczyna, “Generalized kinds of density and the associated ideals,” Acta Math. Hungar., 147, No. 1, 97–115 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  2. G. S. Baranenkov, B. P. Demidovich, V. A. Efimenko, etc., Problems in Mathematical Analysis, Mir, Moscow (1976).

  3. S. Bhunia, P. Das, and S. K. Pal, “Restricting statistical convergence,” Acta Math. Hungar., 134, No. 1-2, 153–161 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  4. V. Bilet, “Geodesic spaces tangent to metric spaces,” Ukr. Math. Zh., 64, No. 9, 1448–1456 (2012); English translation: Ukr. Math. J., 64, No. 9, 1273–1281 (2013).

  5. V. Bilet and O. Dovgoshey, “Isometric embeddings of pretangent spaces in E n,” Bull. Belg. Math. Soc. Simon Stevin, 20, 91–110 (2013).

    MathSciNet  MATH  Google Scholar 

  6. R. Colak, “Statistical convergence of order α,” in: Modern Methods in Analysis and Its Applications, Anamaya Publ., New Delhi, India (2010), pp. 121–129.

  7. J. Connor, “The statistical and and strong p-Cesaro convergence of sequences,” Analysis, 8, 207–212 (1998).

    MathSciNet  Google Scholar 

  8. P. Das, “Certain types of open covers and selection principles using ideals,” Houston J. Math., 39, No. 2, 637–650 (2013).

    MathSciNet  MATH  Google Scholar 

  9. P. Das, “Some further results on ideal convergence in topological spaces,” Topol. Appl., 159, 2621–2625 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Das and D. Chandra, “Some further results on ℐ−γ and ℐ−γk-covers,” Topol. Appl., 16, 2401–2410 (2013).

  11. P. Das and S. K. Ghosal, “On ℐ-Cauchy nets and completeness,” Topol. Appl., 157, 1152–1156 (2010).

  12. P. Das and S. K. Ghosal, “When ℐ-Cauchy nets in complete uniform spaces are I-convergent,” Topol. Appl., 158, 1529–1533 (2011).

  13. P. Das and E. Savas, “Some further results on ideal summability of nets in (ℓ)-groups,” Positivity, 19, No. 1, 53–63 (2015).

  14. O. Dovgoshey, “Tangent spaces to metric spaces and to their subspaces,” Ukr. Mat. Visn., 5, 468–485 (2008).

  15. O. Dovgoshey, F. G. Abdullayev, and M. Kücükaslan, “Compactness and boundedness of tangent spaces to metric spaces,” Beitr. Algebra Geom., 51, 100–113 (2010).

  16. O. Dovgoshey and O. Martio, “Tangent spaces to metric spaces,” Rep. Math. Helsinki Univ., 480 (2008).

  17. I. Farah, “Analytic quotients. Theory of lifting for quotients over analytic ideals on integers,” Mem. Amer. Math. Soc., 148 (2000).

  18. H. Fast, “Sur la convergence statistique,” Colloq. Math., 2, 41–44 (1951).

    MathSciNet  MATH  Google Scholar 

  19. A. R. Freedman and J. J. Sember, “On summing sequences of 0’s and 1’s,” Rocky Mountain J. Math., 11, 419–425 (1981).

  20. J. A. Fridy, “On statistical convergence,” Analysis, 5, 301–313 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Heinonen, Lectures on Analysis on Metric Spaces, Springer (2001).

  22. P. Kostyrko, T. Šalát, and W. Wilczyński, “ℐ-convergence,” Real Anal. Exchange, 26, 669–685 (2000/2001).

  23. M. Kücuükaslan, U. Deger, and O. Dovgoshey, “On the statistical convergence of metric-valued sequences,” Ukr. Math. Zh., 66, No. 5, 796–805 (2014); English translation: Ukr. Math. J., 66, No. 5, 712–720 (2014).

  24. B. K. Lahiri and P. Das, “ℐ and ℐ*-convergence in topological spaces,” Math. Bohem., 130, 153–160 (2005).

    MathSciNet  Google Scholar 

  25. M. Mačaj and T. Šalát, “Statistical convergence of subsequences of a given sequence,” Math. Bohem., 126, 191–208 (2001).

    MathSciNet  MATH  Google Scholar 

  26. H. I. Miller, “A measure theoretic subsequence characterization of statistical convergence,” Trans. Amer. Math. Soc., 347, 1811–1819 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Papadopoulos, “Metric spaces, convexity and nonpositive curvature,” Eur. Math. Soc. (2005).

  28. T. Šalát, “On statistically convergent sequences of real numbers,” Math. Slovaca, 30, 139–150 (1980).

    MathSciNet  MATH  Google Scholar 

  29. H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloq. Math., 2, 73–74 (1951).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Jadavpur University, West Bengal, India

    P. Das

  2. Istanbul Commerce University, Istanbul, Turkey

    E. Savas

Authors
  1. P. Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Savas
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 12, pp. 1598–1606, December, 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Das, P., Savas, E. On Generalized Statistical and Ideal Convergence of Metric-Valued Sequences. Ukr Math J 68, 1849–1859 (2017). https://doi.org/10.1007/s11253-017-1333-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-017-1333-7

Search

Navigation